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A homogenization method used to predict the
performance of silencers containing parallel splitters
Benoit Nennig, Rémy Binois, Emmanuel Perrey-Debain, Nicolas Dauchez
To cite this version:
Benoit Nennig, Rémy Binois, Emmanuel Perrey-Debain, Nicolas Dauchez. A homogenization method
used to predict the performance of silencers containing parallel splitters. Journal of the Acoustical
Society of America, Acoustical Society of America, 2015, 137 (6), pp.11. �10.1121/1.4921598�.
�hal-01178969�
A homogenization method used to predict the performance of silencers
containing parallel splitters
Benoit Nenniga)and Remy Binois
Laboratoire d’Ing´enierie des Syst`emes M´ecaniques et des MAt´eriaux, (LISMMA-QUARTZ EA2336), SUPMECA, 3 Rue Fernand Hainaut, 93407 Saint-Ouen Cedex, France.
Emmanuel Perrey-Debain and Nicolas Dauchez
Sorbonne universit´es, Universit´e de Technologie de Compi`egne,
Laboratoire Roberval, UMR CNRS 7337, CS 60319, 60203 Compi`egne cedex, France. (Dated: July 22, 2015.)
An analytical model based on a homogenization process is used to predict and understand the behavior of finite length splitter/baffle-type silencers inserted axially into a rigid rectangular duct. Such silencers consist of a succession of parallel baffles made of porous material and airways inserted axially into a rigid duct. The pore network of the porous material in the baffle and the larger pores due to the airway can be considered as a double porosity (DP) medium with well-separated pore sizes. This scale separation leads by homogenization to the DP model, widely used in the porous material community. This alternative approach based on a homogenization process sheds physical insight into the attenuation mechanisms taking place in the silencer. Numerical comparisons with a reference method are used to show that the theory provides good results as long as the pressure wave in the silencer airways propagates as a plane wave parallel to the duct axis. The explicit expression of the axial wavenumber in the DP medium is used to derive an explicit expression for the optimal resistivity value of the porous material, ensuring the best dissipation for a given silencer geometry.
PACS numbers: 43.20.Mv, 43.50.Gf, 43.28.Py
Keywords: Silencer, Muffler, Baffles, Splitters, Absorbing materials, Porous materials, Effective medium, Double porosity materials, Design
I. INTRODUCTION
Baffle and splitter-type silencers are widely used in air conditioning systems in buildings to reduce the noise emitted by air-moving devices such as fans. They consist of a periodic succession of parallel baffles made of porous material (usually mineral wool) and airways inserted ax-ially into a rigid duct acting as a waveguide.
Classical methods used to compute the propagation of acoustic waves through such silencers are generally based on modal techniques, such as the well-known mode matching method. This method relies on solv-ing the appropriate eigenproblem over the cross-section of the duct1–7. Eigenfunctions can be solved either
analytically5, using a root finding algorithm3,6 or by discretization techniques using trigonometric function decomposition1,2, the finite element method4 or layered
discretizations7. It is also possible to perform
three-dimensional finite element analysis8,9. A comparison
be-tween some of these methods can be found in the recent review of Kirby8 on modeling automotive dissipative
si-lencers.
In many practical configurations1,4, the frequency
range of interest is low enough to allow focusing on the fundamental mode carrying the acoustic energy in the silencer. We will call this type of wave propagation the plane wave regime as the modal profile is almost
con-a)Electronic address: [email protected]
stant over the airway cross-section. In this situation, the combination of air and porous layers can be considered as a double porosity medium composed of the pore net-work of the porous material in the baffle and the larger pores due to the airway. This homogenization approach leads to an explicit expression of the axial wavenumber for the plane wave mode, which greatly facilitates ana-lyzing the performance of this type of silencer. To the authors’ knowledge, no attempt has been made to use such an approach for silencer design.
The double porosity (DP) models introduced in au-dible acoustics by Auriault et al.10 and Olny et al.11,
have been used widely to model the absorption of per-forated porous media11–14, multiscale porous media such
as porous granular beads15 and media involving the
in-clusion of one porous material in another one16. This
concept has given rise to innovations and contributed to the success of heterogeneous materials used for acoustic absorption, especially in the low frequency regime12. All
these studies were limited to rigid frame porous media17
((see Chap. 5)), though Dazel et al.14 has recently
pro-posed a DP model taking into account skeleton elasticity effects.
The main objectives of this paper are: (i) to illustrate the interest and efficiency of DP formalism to model si-lencers in the plane wave regime; and (ii) to exploit the explicit expression of the wavenumber in the silencer to derive design rules.
The paper is organized as follows. First, the principles of the transfer matrix method (TMM) and the DP model are recalled and combined to obtain a simple baffle-type
Ls
H Incident plane wave
macroscopic scale Elementary cell of DP mesoscopic scale
b 2a
Double porosity media (Homogeneous effective material)
I II III
Porous material / Baffle
microscopic scale
z y
Air pore / Airway
FIG. 1. Geometry of the silencer.
silencer model. The results are shown in terms of Trans-mission Loss (TL) and compared with reference numer-ical results associated with the exact geometry of the silencer. Finally, the DP model is used to derive an opti-mal value for the resistivity yielding the best dissipation for a given silencer geometry.
II. DOUBLE POROSITY BASED SILENCER MODEL A. Problem statement
The bidimensional silencer considered here and sketched in Fig. 1 consists of a periodic succession of par-allel baffles made of porous material and airways. Semi-infinite ducts are present on each side of the silencer (re-gion II), at the inlet (re(re-gion I) and the outlet (re(re-gion III). It is assumed that all the regions have the same cross-section H and rigid walls. Let N be the number of baffles, φp = a/(a + b) the open area ratio and ¯φp = 1 − φp its
complement, i.e. the filling fraction. These quantities are linked by H = 2(a + b)N , thus changing the num-ber of baffles implies that the baffle thickness 2b must be modified accordingly (H and φp are kept fixed).
In each region, only the fundamental mode is allowed to propagate, thus the acoustic pressure field p fulfills the one-dimensional Helmholtz equation (eiωt) along
di-rection z which corresponds to the axial coordinate: ∂z2pi+ ki2pi= 0, (1)
with wavenumber ki. Subscript i = 0 is used in regions I
and III and subscript i = dp in region II. Note that the convection effect is neglected, the Mach number being less than 0.05 for typical buildings applications.
For a simple duct element of length L, pressure p and axial velocity v at the beginning (b) of the section are related to pressure p and axial velocity v at its end (e)
by p v b = cos kiL iZisin kiL
(i/Zi) sin kiL cos kiL
| {z } T p v e (2)
where Zi = ρici, ρi and ci are the characteristic
impedance, the density and the celerity, respectively, in region i. Note that here i = dp as it corresponds to the silencer region. To make the presentation as simple as possible, there are no rigid fairings or perfo-rated plate separating the air domain from the porous media. Under these conditions and for the fundamen-tal mode excitation in region I, the transmission coeffi-cient and the backward reflection coefficoeffi-cient are given by T = 1/M11 and R = M21/M11, respectively. Here the
matrix M = B−1T B, with B = 1 1 1/Z0 −1/Z0 , (3)
links the reflected and incident waves. It should be noted that the transfer matrix T can be modified easily to take into account the presence of a perforated plate on both sides of the silencer, by multiplying T by the appropriate discontinuity matrices18,19. Finally, the transmission loss
of the silencer is given by
TL = −20 log10|T |. (4)
In principle, the previous analysis only applies below cut-off. In fact, above the cut-off frequency of the first trans-verse mode (fc = c0/(2H)), the analysis still holds as
long as the incident, reflected and transmitted pressure fields remain plane. In the present configuration, the periodic and symmetric arrangements of baffles implies that the indexes of the modes propagating through the silencer must satisfy the selection rule1 (see (A5) in
Ap-pendix A):
n = ninc+ 2qN. (5)
Here, ninc is the order of the incident mode and q is a relative integer. In the present context, ninc = 0 as
only the plane wave mode is considered. It follows that when N = 1, only even modes are allowed to propagate. If N = 3, only modes of order n = 6q are allowed to propagate, the other modes being forbidden, etc. The selection rules indicates that the wave propagation in the silencer remains in the plane wave regime as long as the frequency is below the cut-off frequency of the mode of order 2N .
This propagation model is very simple, but the diffi-culty now is to determine the effective wavenumber and the characteristic impedance of region II. This is the sub-ject of the next section.
B. Double porosity material
The acoustics of the double porosity material were es-tablished previously by Auriault and Boutin10using peri-odic structure homogenization techniques. The approach
was extended by Olny et al.11 to clarify the influence,
at the macroscale, of contrasting permeability occurring between the macro- and micro-pores of sound absorbing materials. DP models were used to enhance the normal incidence sound absorption of porous materials by Atalla et al.20 and Sgard et al.12. In the following, the
princi-ples of the DP material model are recalled briefly, using the formalism of Refs. 11, 12, 17 for slit perforations.
Three characteristic lengths are required to describe each scale of the DP material. At macroscopic scale, the characteristic length ℓ is governed by the wavelength λ such as ℓ ∼ λ/2π. At the mesoscopic scale, ℓp is of the
same order as the size of the air gap between the baffle. Finally, at the microscopic scale ℓm is of the same order
as the radii of the pores of the porous material. The subscripts p, m and dp are used throughout the paper for the pores (airways), the micropores (porous material) and the double porosity medium (silencer), respectively. To ensure the separation of scale and apply the pe-riodic structure homogenization method, it is necessary that ℓm ≪ ℓp ≪ ℓ. The high permeability contrast
as-sumption will be used throughout this paper. This means that the resistivity of the airway and that of the micro-porous medium are very different. Therefore, their pore sizes are also different and ℓm/ℓp< 10−2. This
assump-tion is not restrictive as it allows recovering the low per-meability contrast model11. The range of validity of the
DP model depends on the frequency, the porous material and the geometry.
An additional assumption concerns the length of the silencer L which must be much larger than the meso-scopic length scale ℓp so that both ends of the silencer
have negligible effects on the diffusion mechanisms tak-ing place in the double-porosity material. On the ba-sis of these assumptions, Olny et al.11 showed that the
macroscopic behavior of acoustic waves is given by the Helmholtz equation (1) with the effective wavenumber as follows:
kdp= ω
q
ρdp/Kdp, (6)
involving an effective density ρdp and an effective bulk
modulus Kdp. The details of such parameters are now
listed for air slits of width 2a between two layers of porous material of width b (the results are also available in Ref. 11 for circular holes). With the dynamic viscosity denoted as η, the effective density can be written as:
ρdp=
η iωΠdp
, (7)
where Πdp is the dynamic permeability in direction z:
Πdp = ¯φpΠm+ Πp. (8)
Here, Πm can be deduced from (7) and (B1) by simply
replacing the subscript dp by m. The effective density ρm
and the effective bulk modulus Kmof the porous material
are given by the Johnson-Champoux-Allard model, given in appendixB. The dynamic permeability in the meso-pore is given by
Πp = −iφpδ2vF (µv), (9)
involving the function
F (µ) = 1 − tanh µ √ i µ√i ! . (10)
Here, µv = a/δv is the ratio between the air gap
width and the viscous boundary layer thickness δv =
p
η/(ρ0ω).
Let us now introduce the bulk modulus in the airways Kp, obtained using the simplified Lafarge model21
Kp= γP0/φp γ − i(γ − 1) Θp δ2 tφp , (11)
with thermal permeability
Θp= −iφpδt2F (µt), (12)
the thermal boundary layer thickness δt=pκ/(ρ0Cpω)
and the ratio µt = a/δt. Because the viscous and
ther-mal boundary layers are much sther-maller than the airway area, the quantities µt and µv are large enough for us
to reasonably assume that F (µt) ≈ F (µv) ≈ 1, meaning
that Kp≈ γP0/φp and Πp ≈ −iφpη/(ρ0ω), (13) and therefore ρdp ≈ φp ρ0 + φ¯p ρm −1 . (14)
The bulk modulus of the DP material is a combination of the bulk modulus of the micro-porous media Kmgiven
explicitly in (B2) and Kp, giving
Kdp= 1 Kp + ¯φp Fd ω P0 φmKm Km −1 , (15) where Fd(ω) = 1 − iω ωd D(ω) D(0), (16)
is a function that links the mean pressure in the micro-pore (i.e. in the baffle) to the average pressure in the airway. For the slits, Olny et al.11gave D
0= ¯φpb2/3 and
D(ω) = −i ¯φpδd2F (µd), (17)
with the ratio µd = b/δd and the pressure diffusion skin
depth δd=
q
P0
σφmω. This parameter gives an estimation
of the boundary layer thickness in the material in which strong pressure gradients are taking place. The diffusion frequency is defined as ωd= 3P0 b2σφ m , (18)
so that Fd has a simpler form quite similar to (10)
Fd(ω) = tanh µd √ i µd √ i . (19)
0 2 4 6 8 10 −0.5 0 0.5 1 Fd ω ωd
FIG. 2. Evolution of Fd(ω), the ratio between the mean
pressure in the micro-porous material and the pore pressure. |Fd|, ImFd, ReFdas given in (16).
The imaginary part of this function reaches a minimum when ω ≈ ωd, as shown in Fig. 2.
Going back to (6) with (7) and (15), it can be shown that the effective wavenumber for the double porosity medium takes the following form:
kdp2 = φpk 2 0 ρ0 + ¯φpFd k2 m ρm φp ρ0 + ¯ φp ρm . (20)
Here we have used the fact that the wavenumber in the air is given by k0= ω/
p
γP0/ρ0. This expression shows
that the effective wavenumber resembles a weighted aver-age of the airway wavenumber k0 and the micro-porous
wavenumber km. This is not an exact rule of mixture
due to the function Fd which is nearly equal to unity
only when ω ≪ ωd.
Before we end this section, it is instructive to observe that function Fd can be obtained by another approach.
By introducing symmetry arguments, the fundamental mode for the pressure in the material takes the approxi-mate form:
p ≈ pp
cos(kmy)
cos(kmb)
e−iβz. (21)
Here, line y = 0 corresponds to the central axis of the baf-fle and pp is the air pressure at interface y = b. This is a
reasonable assumption at a sufficiently low frequency, so it is reasonable to assume that the transverse wavenum-ber is much larger than the axial one β, i.e. km ≫ β.
The material behaves as if it were reacting locally while the pore pressure behaves like a forcing term. In these circumstances, it is clear that
hpmi
pp ≈
tan kmb
kmb
. (22)
By definition, this quantity is identical to function Fd in
(19), thus, after identification: km≈√ω ·
r −iφPmσ
0
. (23)
This expression is precisely the low frequency approxi-mation of the wavenumber in the porous material: km=
ωpρmKm−1(it sufficient to consider the low frequency
be-havior of ρm and Km given in the Appendix). The fact
that km increases as the square root of the frequency
with the equal real and imaginary parts (remember that √
−i = (1 − i)/√2) is typical of the viscous regime. It corresponds to the low-frequency range: ω ≪ ωb where
ωb= σφm
ρ0α∞
(24) is the Biot frequency and when ω ≈ ωb, the inertia
forces are of the same order of magnitude as the viscous forces17.
C. Comparison with a reference solution
In this section the TMM model is compared with a mode matching method developed by the authors in Ref. 7. The model takes into account the existence of evanescent acoustic modes at both ends of the si-lencer and converges to very accurate solutions as long as the discretization is sufficiently refined. The results are shown for the TL in Fig. 3. The silencer of length LS = 0.3 m is inserted in a main duct of height H =
0.20 m, and in all cases the airway area is kept constant φp = 0.50 irrespective of the number N of baffles. It is
clear that the TMM combined with the DP model (eq. (20)) provides good to excellent agreements, especially in the case of a large number of baffles (N = 5 here). With wool GW1 of moderate resistivity, the differences never exceed 3 dB. The oscillations in the TL curve at low frequency with wool GW2 and with one baffle re-semble those of an expansion chamber. This effect is due to the high resistivity of the micro-porous material (its properties are reported in Tab. I).
With both wools, when N = 1, discrepancies are very noticeable around 1700 Hz which corresponds to the cut-off frequency of the first even mode in the rigid duct. When N = 3 or 5 baffles, the selection rule (A5) indicates that the problem remains essentially one-dimensional even above cut-off. In this regime the DP model remains valid and is expected to produce excellent results over a wide frequency range.
We will now take advantage of the full numerical model7 to describe in greater detail the acoustic
pres-sure pattern in the silencer and its correspondence with the diffusion frequency ωd. This frequency separates the
behavior of the DP medium into three regimes. In the low frequency regime, i.e. ω ≪ ωd, the pressure is
al-most uniform over the silencer cross-section. This is il-lustrated in Fig. 4a when ω = ωd/3. When ω approaches
ωd, the wavelength in the micro-porous material is
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 5 10 15 20 25 f (Hz) TL (dB) 5 1 (a) GW1 wool (σ = 14 066 Nm-4s) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 5 10 15 20 25 f (Hz) TL (dB) 5 1 (b) GW2 wool (σ = 135 000 Nm-4s)
FIG. 3. Comparison between TL prediction given by a refer-ence method7( ) and the DP model ( ) for 1, 3 and 5
baffles in a silencer such H = 0.2 m and LS= 0.3 m. a) with
the GW1 wool, b) with the GW2 wool.
pressure in the core of the baffle is about half the pressure in the airway (see Fig. 4b). Both pore networks (micro-porous and the airways) are strongly coupled. The pres-sure fields in each domain present a phase mismatch, gen-erating a new dissipation effect that does not exist in a simple porosity medium11, leading to higher attenuation.
This effect can be identified in Fig. 3b and for one baf-fle. In this case, the diffusion frequency fd = 152 Hz is
sufficiently low to be discernible on the TL curve (Table II gives the Biot and diffusion frequencies for different configurations). Finally, when ω ≫ ωd, the pressure
van-ishes inside the baffle except on a thin boundary layer of thickness δd and the core of the baffle has little
influ-FIG. 4. Normalized pressure field (in absolute value) below, around and above ωdfor a GW2 foam in a silencer such H =
0.2 m, LS = 0.3 m and N = 1. Obtained with a reference
solution7.
TABLE I. Material properties used in numerical tests. With the porosity φm, flow resistivity σ, the tortuosity α∞, the
viscous and thermal characteristic lengths Λ and Λ′
. Material φm σ α∞ Λ Λ ′ Ref. - [Nm-4s] - [µm] [µm] -GW1 0.954 14 066 1.0 91.2 182.4 [7] GW2 0.94 135 000 2.1 49 166 [12]
ence (see Fig. 4c). It can be seen that the pressure inside the airway remains nearly constant over the cross-section area for each case. Knowing ωd is useful to understand
the different dynamical regimes of the silencer as it is strongly linked to its design.
TABLE II. Characteristic frequencies for 1, 3 and 5 baffles with φp= 0.5 and H = 0.2 m.
Material fb= ω2πb (Hz) fd= ω2πd (Hz)
1 3 5
GW1 1 760 1 442 12 979 36 053
III. DERIVATION OF OPTIMAL PARAMETERS
In the previous section the interest of combining the TMM and DP models was shown. This simple approach represented a fast and reliable tool for designing baffle silencers. The aim of this section is to establish guide-lines for designing silencers with the best attenuation as a function of the frequency range of interest. Fol-lowing the one-dimensional analysis, best attenuation is achieved my minimizing the magnitude of the transmit-ted coefficient T = 1 M11 = 1 cos(kdpL) + izssin(kdpL) (25) where zs= 1 2 Zdp Z0 + Z0 Zdp (26) can be interpreted as an impedance mis-match coefficient for the silencer. The importance of this term depends mainly on the open area ratio φp since, in the low
fre-quency limit, it can be shown that lim ω→0 Zdp Z0 = q 1 φ2 p+ γφpφ¯pφm . (27)
In practical situations, 25% < φp < 75% and the
magnitude of Zdp never exceeds twice the
characteris-tic impedance Z0. Thus, we can reasonably assume that
zs= 1 + ε where ε < 0.3 and consequently
T = exp(−ikdpL)(1 + O(ε)). (28)
This signifies that the Transmission Loss is mainly driven by the imaginary part of the wavenumber, i.e. Imkdp. The explicit expression for the wavenumber kdp in (20), derived on the basis of physical arguments, allows understanding the role of certain parameters, especially the resistivity of the porous material and the open area ratio, which are clearly the most influential. Therefore, it is natural to identify its dependence on these two pa-rameters. In the following sections, the quantity Im kdp
must be considered as a function of ω, σ and φp. The
dimensionless parameters rΛ and c corresponding to the
ratio of the characteristic lengths and the square root of the shape factor (see Appendix) respectively, are kept constant in all the calculations, while the thermal and viscous lengths must be considered as functions of σ to ensure realistic conditions22.
A. Low frequency approximation
We will assume that the frequency is sufficiently small so that the two dimensionless quantities ǫd = ω/ωd and
ǫb = ω/ωb are small compared to unity: ǫd ≪ 1 and
ǫb ≪ 1. In all cases, low-frequency approximations will
be identified by the symbol“ ˜ ”. To the first order, the function Fd has the simple form
˜
Fd(ω) = 1 − iǫd. (29)
Then using the low frequency approximation of ˜Kmgiven
in (B5), we have ˜ Kdp= γP0 φp+ γ ¯φpφm(1 − i(ǫd+ Υǫb)) −1 . (30)
Finally, replacing the low frequency approximation ˜ρm
in (14) yields the approximated expression for the DP wavenumber: ˜ k2dp= k2 0 φp · φp+ γ ¯φpφm(1 − i(ǫd+ Υǫb)) 1 + φmφ¯p φpα∞iǫb . (31)
A closed form expression for the imaginary part of the DP axial wavenumber can be derived by calculating, to the first order,
Re ˜k2dp= k 2 0 φp φp+ γ ¯φpφm , (32a) Im ˜k2dp= −k02φ¯pφ φp ǫdγ + ǫb γΥ + 1 α∞ +γ ¯φpφm α∞φp . (32b) which means that Im ˜kdp is of order ǫd or ǫb and
Im ˜kdp =1 2 Im ˜k2 dp q Re ˜k2 dp (33) is a good approximation. In this form, the imaginary part of ˜kdp can be easily differentiated with respect
to σ and the location of the extremum σ∗
0, satisfying
∂σ(Im ˜kdp)(σ∗0) = 0 can be found by solving
∂ǫd ∂σγ + ∂ǫb ∂σ γΥ + 1 α∞ +γ ¯φpφm α∞φp = 0. (34)
The solution to this equation is unique and does not de-pend on the frequency:
σ∗ 0= σdb s 1 γα∞ +φ¯pφm φpα∞ + Υ . (35)
The prefactor corresponds to σdb, the value taken by the
resistivity when the diffusion and Biot frequencies are equal (ωb= ωd), σdb= s 3P0ρ0α∞ b2φ2 m . (36)
Physically, this value stems from a compromise between pressure diffusion effects (ωd ∝ 1/σ) and visco-thermal
effects (ωb ∝ σ). It is interesting to evaluate the order
of magnitude of each term in the square root of (35) for low tortuosity: 1/γ ≈ 0.7 and Υ ≈ 1.3. The second term, which involves the ratio between the filling fraction and the open area ratio, is not bounded, although in practice it is reasonable to assume that 1/4 ≤ φp≤ 3/4, meaning
that φ¯pφm
φp ∈ [0.3, 3]. Thus we can already anticipate that
in many configurations of practical interest, the optimal value for the resistivity should be chosen in the approxi-mate range: 1.5σdb≤ σ∗0 ≤ 2.5σdb. It is noteworthy that
this optimal value involves two geometric parameters, the first one, φp, is dimensionless whereas the height of the
B. Low resistivity and low frequency approximation
The previous analysis holds as long as the resistivity σ is not vanishingly small. Otherwise, ǫbcannot be treated
as a very small parameter. Although very low resistivity materials (i.e. below 5000 Nm-4s) are seldom
encoun-tered in practice, it is interesting to identify the exis-tence of another optimal value for resistivity when both the frequency and resistivity are low. To make the prob-lem tractable, we will set ǫdto zero so Fd≈ 1 and assume
that ǫblies somewhere in the region ǫb∈ [0.1, 1].
Contin-uing with these assumptions, closer analysis reveals that the DP wavenumber can be reasonably approximated by
˜ k2 dp = k02 φp+ γ ¯φp/φm φp+ ¯ φp α∞ φm(r−i ωb ω) . (37)
The extremum problem ∂σ(Im ˜kdp) = 0 using (33) can
be resolved analytically and we find that the solution depends linearly on the frequency as
σ∗ 1(ω) = ω · ρ0α∞ φmφp (rφp) 1 4 φm ¯ φp α∞ + rφp 3 4 . (38)
Unlike the previous case, the thickness of the baffle b is not involved and formula (38) remains unchanged if the number of baffles is modified while keeping φp constant.
C. Discussion
1. Validity of Eqs. (35) and (38)
The case of a typical baffle silencer is now analyzed and discussed for 3 values of the open area fraction φp: 0.25,
0.5 and 0.75. In every case, the silencer is composed of one baffle based on GW1 wool (rΛ, c, α∞, φm) and
the height of the duct is H = 0.2 m. In Figs. 5, the attenuation rate in the DP material, i.e. the imaginary part of the wavenumber kdp normalized by k0, is plotted
as a function of the frequency and resistivity. It should be noted that the best acoustic attenuations occur in the white area, the extent of which depends strongly on the open area ratio and, as expected, the larger value of φp
is, the worse the silencer attenuation.
The maximum (with respect to σ) attenuation rate given by the DP model is also shown and these optimal values are compared with those calculated with the ref-erence model7. For the latter, the axial wavenumber β
corresponding to the fundamental mode is considered. The locations of these maximums are in good agreement though discrepancies may occur in regions where Im kdp
is nearly constant with respect to the resistivity. When this happens, a jump from one isoline to the other can be observed, as shown in Fig. 6. The frequency of this jump depends mainly on the open area ratio. At low frequency, the best attenuation closely follows the theo-retical predictions given by σ∗
1(ω) in (38) and above the
jump frequency the value for the resistivity σ∗
0 in (35)
can be observed to be nearly optimal.
0.2 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 f (Hz) σ (kNsm−4) 200 400 600 800 1000 1200 1400 1600 1800 2000 10 20 30 (a) φp= 0.25 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 f (Hz) σ (kNsm−4) 200 400 600 800 1000 1200 1400 1600 1800 2000 10 20 30 (b) φp = 0.50 0.03 0.03 0.03 0.03 0.03 0.07 0.07 0.07 0.07 0.07 0.1 0.1 0.1 f (Hz) σ (kNsm−4) 200 400 600 800 1000 1200 1400 1600 1800 2000 10 20 30 40 50 (c) φp= 0.75
FIG. 5. Influence of σ on the attenuation rate: isovalues of Imkdp/k0 ( ) for φp = 0.25, 0.5 and 0.75. The markers
stand for the maximum (with respect to σ) of the attenuation rate given by the DP model (max Im kdp/k0 : +) and by the
reference model7(max Im β/k
0: ◦). The optimal values given
by Eqs.(35) and (38) are represented by dotted-dashed lines ( ). Silencer with H = 0.2 m and one baffle.
200 400 600 800 1000 1200 1400 1600 1800 2000 10 20 30 f (Hz) σ (kNsm−4) σ∗ 0 σ∗ 1 Jump
FIG. 6. Root locus of ∂σ(Im kdp(ω)) = 0 ( )
computed with finite difference. The makers stand for the maximum of the attenuation rate given by the DP model (max Im kdp/k0 : +) and a reference solution mode
computation7 (max Im β/k
0 : ◦). The optimal values given
by Eqs.(35) and (38) are represented by dotted-dashed lines ( ). Silencer with H = 0.2 m, φp= 0.5 and one baffle.
TABLE III. Optimal resistivity value for various configura-tions. φp H N (rΛ, c, α∞, φm) σ ∗ 0 f ∗ d = ω∗ d 2π - [m] - based on [Nm-4s] [Hz] 0.25 0.2 1 GW1 17 375 518 0.50 0.2 1 GW1 19 238 1 054 0.75 0.2 1 GW1 32 683 2 483 0.75 0.7 1 GW1 9 338 709 0.50 0.7 1 GW1 5 497 301 0.50 0.7 3 GW1 16 490 904
By performing intensive numerical tests involving dif-ferent silencer geometries and porous materials with low to moderate tortuosity (between 1 and 1.5), it was found that (35) provides a very good estimation of the optimal resistivity, except in the very low frequency regime. For illustrative purposes, Table III gives the values of σ∗
0 for
various configurations. The latter correspond to avail-able realistic materials. The last column gives the cor-responding diffusion frequency ω∗
d = 3P0/(b2σ0∗φm). At
this frequency the two predictions Eqs. (35) and (38) are nearly equal and the attenuation rate reaches its maxi-mum value, as confirmed in Figs. 5.
2. Practical examples
In order to illustrate the practical use of the optimal value for the resistivity σ∗
0, we consider a typical
baf-fle silencer used in ventilation systems4 with L
s = 1.2
m and H = 0.7 m. The silencer comprises N = 1 or N = 3 baffles. The parameters of the porous material
are based on those of the GW1 wool (rΛ, c, α∞, φm)
ex-cept that the resistivity can be modified independently. Fig. 7 shows the Transmission Losses of the silencer for three scenarios. The results were computed with the ref-erence model7 and thus can be considered as exact. In
particular they take into account reflection mechanisms which are ignored in the derivation of the optimal resis-tivity. Note that the Mechel selection rule from (5) was used to identify the plane wave regime in order to set the frequency range of interest accordingly (around 500 Hz for N = 1 and 1500 Hz for N = 3). The performances of the silencer can be changed significantly by varying the resistivity from σ∗
0/10 to 10σ∗0 artificially. It is
note-worthy that in each case, choosing σ = σ∗
0 yields the
best attenuation. As expected, this choice is not optimal at low frequencies but the drop in performance is rather marginal. Finally, the DP model, which is an approxi-mate and homogenized representation of the real silencer, is used in conjunction with the TMM in order to obtain an estimate of the TL for σ = σ∗
0. The curve shows
some-what good agreement, especially at low frequencies, and deviates strongly once the conditions for the plane wave regime in the silencer are no longer satisfied.
Previous results are valid only in the plane wave regime, assuming and incident plane wave with no mode conversion. However, it is interesting to test the validity of the optimal resistivity when higher mode are imping-ing the silencer. In this regard, two standard excitations are usually adopted23 : i) one assumes equal energy per
mode or ii) equal energy density per mode (in this case the amplitudes of normalized propagating modes are all equal). Simulation with equal energy density per propa-gating mode are performed with a reference method on configurations (b) and (c) taken from Fig. 7. It is shown in Fig. 8 that, even with equal energy density per mode excitation, the optimal resistivity given in Eq. (35) pro-vides the best compromise over a large frequency interval and allows achieving nearly the best attenuation as long as there is no modal conversion among incident modes, i.e. when the frequency is below the cut-off frequency given by the selection rule. For instance around 500 Hz for N = 1 in Fig. 8a and around 1500 Hz for N = 3 in Fig. 8b. Note that other calculations assuming equal energy per mode excitation leads to very similar results.
IV. CONCLUSION
In this paper a double porosity model was used to pre-dict and optimize the acoustic performances of baffle-type silencers in ducts. The model, originally developed to describe the wave propagation in porous materials with slit-like perforations, is based on a homogenization process and thus yields an explicit expression for the ef-fective wavenumber in the silencer. Through numerical comparisons, it was shown that the theory provides a good estimate as long as the pressure wave in the silencer propagates like a plane wave parallel to the duct axis.
In the last section of the paper, the analytical formula for the wavenumber was fully exploited to predict the conditions under which the best acoustic performances
0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 12 14 16 18 f (Hz) TL (dB) DP (a) φp= 0.75, N = 1 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25 30 f (Hz) TL (dB) DP (b) φp= 0.5, N = 1 0 500 1000 1500 0 10 20 30 40 50 60 70 80 90 f (Hz) TL (dB) DP (c) φp= 0.5, N = 3
FIG. 7. Transmission Loss evolution for σ ∈ [0.1σ∗
0, . . . , σ
∗
0, . . . ,10σ
∗
0] computed with a reference method7 for a plane wave excitation. If σ < σ∗
0, dotted line
are used (. . .). If σ > σ∗
0, dash-dotted line are used ( ).
The grayscale tends to the darkest value when the resistivity tends to σ∗
0. If σ = σ ∗
0, black is used. The bold continuous
line ( ) stands for the reference solution and the dashed line ( ) stands for the DP model with TMM for σ = σ∗
0.
Silencer with H = 0.70 m and Ls = 1.20 m, based on wool
GW1. 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 f (Hz) TL (dB) (a) φp= 0.5, N = 1 0 500 1000 1500 2000 2500 0 10 20 30 40 50 60 70 f (Hz) TL (dB) (b) φp= 0.5, N = 3
FIG. 8. Transmission Loss evolution for σ ∈ [0.1σ∗
0, . . . , σ
∗
0, . . . ,10σ
∗
0] computed with a reference method7 with Equal energy density per mode excitation
case23. Silencer with H = 0.70 m and L
s= 1.20 m, based on
wool GW1. Same legend as for Fig. 7
can be obtained. It turned out that: (i) at low frequency, the optimal resistivity given by (38) should vary linearly with the frequency; (ii) above a certain frequency, the value for the resistivity in Eq. (35) should be used. In practice, numerical predictions in terms of Transmission Losses show that Eq. (35) is always nearly optimal and that the drop in performance at low frequencies remains marginal.
In principle, the value proposed for the optimal resis-tivity is restricted to baffle-type silencers but there are good reasons to believe that a similar approach could be applied to optimize other types of silencers comprising a succession of porous layers such as expansion cham-bers filled with a porous material. It could also be used to design surface acoustic absorbers illuminated at nor-mal incidence, in the spirit of Ref. 12 for instance. The present study focused on rigid frame materials and it would be interesting to investigate the behavior of limp materials22and poroelastic materials. Convection effects
due to the presence of mean flow could also be the subject for further investigation.
V. ACKNOWLEDGEMENTS
The authors would like to thank their industrial part-ners Alhyange Acoustique
(see http://www.alhyange.com/). This work was partly funded by the ANR Project METAUDIBLE No. ANR-13-BS09-0003-01 funded jointly by ANR and FRAE.
APPENDIX A: SELECTION RULE FOR BAFFLE-TYPE SILENCERS
A complete explanation can be found in Ref.1. Here
we shall recall the main ingredients of the theory. The idea is to view the silencer as an infinite succession of baffles (or equivalently, an infinite repetition of the duct interval y ∈ [0, H]). An incident plane wave of the form
pinc= exp(i(αincy − βz)), (A1) gives rise to a series of reflected plane waves
p =X
q∈Z
Rqexp(i(αqy + βqz)). (A2)
The periodic nature (with period d = H/N ) of the prob-lem implies the Floquet relation
αq = αinc+
2πq
d . (A3)
In the context of guided waves in a symmetric silencer, a duct mode is simply the sum of two plane waves since
2 cos(αqy) exp(−iβqz) = exp(i(αqy − βqz))
+ exp(i(−αqy − βqz)) (A4)
where αq must correspond to the transverse wavenumber
of a duct mode, i.e. αq = nπ/H. The Floquet relation
becomes:
n = ninc+ 2qN, (A5)
which is the selection rule given in Ref.1 which reflects
the two periodicity scales H and d. Equation (A5) holds for all scattered modes including the transmitted ones.
APPENDIX B: RIGID FRAME MODEL
Porous materials with rigid skeleton were well de-scribed by the Johnson-Champoux-Allard (JCA) equiv-alent fluid model17 (Chap. 5). This equivalent fluid has
the equivalent density (e+iωt) ρm=α ∞ρ0 φm h 1 − iωωbGJ(ω) i , (B1)
and the equivalent bulk modulus, Km= γP0/φm γ − (γ − 1) 1 − ic2r2 Λ ωb Pr ω 1 + i 2c2r2 Λ Pr ω ωb 1/2−1. (B2) Here, GJ(ω) = q
1 + ic22ωωb, φm is the porosity, σ is the
flow resistivity, Λ is the viscous length, Λ′
is the thermal length and α∞ is the tortuosity. To express GJ as a
function of ωb, the expression
Λ2= 1 c2
8η ωbρ0
, (B3)
given in Ref. 17 (Eq. (5.25)) is used. The coefficient c, is the square root of the shape factor and is close to unity and c ∈ [0.8, 1.1] (see17 (Tab. 4.1, p. 64)). The
ratio between both characteristic length rΛ = Λ/Λ
′ is
introduced and ranges generally between 0.5 and 0.33. The parameters used in this paper are given in Tab. I.
Moreover, γ is the air specific heat ratio and P0is the
atmospheric pressure, Pr is the Prandtl number and η is the dynamic viscosity and κ is the thermal conductivity of air. If ω ≪ ωb, viscous forces are dominant and the
effective parameters can be replaced by the low-frequency approximations (approximated quantities are denoted by symbol“ ˜ ”): ˜ ρm= ρ0 α∞ φm r − iωb ω , (B4) with r = 1 + c2/4 and ˜ Km= P0 φm 1 + iΥω ωb , (B5) where Υ = γ − 1 γ Pr (rΛc)2 . (B6) In practice, Υ ∈ [1.25, 1.48].
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![FIG. 7. Transmission Loss evolution for σ ∈ [0.1σ 0 ∗ , . . . , σ ∗0 , . . . , 10σ 0 ∗ ] computed with a reference method 7 for a plane wave excitation](https://thumb-eu.123doks.com/thumbv2/123doknet/14447544.518013/10.892.503.814.38.630/transmission-loss-evolution-computed-reference-method-plane-excitation.webp)
